2.1. Fundamental Theory of Random-Evidence Hybrid Reliability Analysis
Let
denote the random variables describing the aleatory uncertainties of model inputs and modeled by probability theory,
represent the epistemic uncertain variables modeled by the evidence theory with BPA,
and
are the number of aleatory uncertain variables and epistemic uncertain variables, respectively. Then, the random-evidence hybrid limit state function can be represented as
. The
th epistemic uncertain variable
can be described by evidence space (
), where
denotes the set of FEs,
is the number of the FEs of
,
represents the corresponding BPA of each FE and
. Then, the evidence space
of epistemic uncertain vector
Y can be calculated by Cartesian product method (CPM). Specifically,
where
represents the
kth JFE of
, and the number of JFEs of
Y is given by
. As shown in
Table 1, consider
Y with two epistemic uncertain variables
and
containing two and three mutually exclusive FEs, respectively. Obviously,
and
.
According to the unified reliability analysis with aleatory and epistemic uncertainties proposed by Du [
31,
32], the failure probability
containing both aleatory uncertain variables and epistemic uncertain variables can be expressed as
Since
Yj can be considered as a vector containing
interval variables,
has its lower and upper bounds, which are denoted as
and
, respectively. Then, the belief measure
Bel(
F) and plausibility measure
Pl(
F) can be calculated as follows according to Equation (3).
where
.
2.2. Brief Introduction to Random-Evidence Hybrid CPF Plot
The random-evidence RSA technique proposed by Li et al. not only can screen out important input variables that have significant impact on
Pl, but also determines which part of the input variable has the greatest impact on
Pl. According to their work [
15], the second-level limit state function of each FE should be firstly established based on the non-probability index. Under the framework of non-probability reliability analysis, for a limit state function,
, the non-probability index is given as [
33]:
In general, the larger the non-probability index is, the more reliable the system is. If holds, it indicates that the system must be safe. If holds, it indicates that failure will unavoidably happen to the system. If holds, whether the system is safe or not is uncertain.
In Equation (6),
and
represent the median and deviation of
, respectively.
and
are the lower bound and upper bound of
and can be calculated as follows:
When the system contains both modeled aleatory uncertain variables and epistemic uncertain variables, the non-probability index must be a function of the aleatory uncertain variables
, i.e.,
. For a given JFE
, the corresponding second-level limit state function based on the
Pl can be established as
By setting the quantile of
as
, the failure plausibility measure for
can be defined as
where
,
is the inverse cumulative distribution function (CDF) of
at
.
is an indicator function and is expressed as
It can be seen from Equation (10) that the multiple integral is estimated in the range of
for all aleatory uncertain variables except for
, of which the range is
. Then, the CFP for the aleatory uncertain variable
is expressed as
It can be seen from Equation (12) that the
reflects the effect of the part of distribution of random variable
, i.e.,
, on the
Pl. For
,
of
, assume
and
,
. The following ratio relationship is established:
Since
holds, Equation (13) is rewritten as the following formulation in this paper,
Equation (14) reflects the reduction of Pl due to any reduced uncertain range of .
When only considering the former
FEs of epistemic uncertain variable
, the total number of JFEs is
. Denote the
JFE of
and the corresponding BPA as
and
, respectively. Then, the
Pl, in which the FEs of
is reduced to the former
k FEs, is expressed as
Then, the CFP for
with the former
k FEs is defined as
Assume that for any
and
satisfying
, the following ratio relationship is established.
The meanings of Equations (16) and (17) are similar to those of Equations (12) and (14), respectively. Next, the Monte Carlo solution for the CFP plot is briefly represented below, and this CFP is also discussed.
- (1)
For each JFE , generate samples of aleatory uncertainty variables according to their joint probability density function (PDF). Based on Equation (9), the corresponding output values are obtained.
- (2)
Calculate the values by Equation (11).
- (3)
Sort the samples of the random variables in ascending order and rename them as , the corresponding values of the indicator function Equation (11) are represented as .
- (4)
The
at
q quantile for the random variable
is estimated as follows according to Equation (12):
- (5)
Taking the former
k FEs of the aleatory uncertain variable
, the corresponding
can be given as follows according to Equation (16).
The key to calculating the CFP is to solve the plausibility failure probability, i.e., Pl. However, according to Equation (5), the calculation of Pl(F) is a nested double-loop process. The outer layer calculates the JFEs of and their corresponding . The inner layer estimates the extreme value of the second-level limit state function with respect to each JFE by using the optimization method. According to steps (1) and (2), optimizations are required for each JEF. Therefore, total optimizations are required in step (2) to calculate the extremum of second-level limit state function to determine the value of . For example, for a model including 5 epistemic uncertain variables with 3 focal elements for each epistemic uncertain variable, the total number of JFEs is 35 = 243, therefore, the construction of such a large-scale surrogate model group is a computationally-intensive and time-consuming process. Moreover, assuming the number of samples, i.e., , is equal to 106 generated in step (1), a total of 243 × 106 optimizations are required in step (2) to calculate the value of . Such a large optimization is a serious burden for engineering applications. The shortcomings discussed above severely hinder the application of this RSA method. In order to deal with the above issues, a RS-MCS procedure and KKTO method are introduced in the following section, and then an efficient method is proposed for the calculation of CFP plot.